DES MOINES, Iowa — Once largely united in resisting the Obama administration’s new health care overhaul, a growing number of Republican governors are now buying into parts of the system as the financial realities of their states’ medical costs begin to counterbalance the fierce election politics of the issue.

This week, Michigan’s Rick Snyder became the sixth GOP governor to propose expanding his state’s health insurance program to cover more low-income residents, in line with the Democratic administration’s strong recommendation. Eleven Republican governors have rejected the idea while a dozen, who have been mostly critical, have not announced a decision.

Although the Democratic president’s re-election last fall cleared the way for providing health insurance for millions of Americans who don’t have it, many Republican governors have resisted parts of the plan that remained optional. They have been reluctant to expand their Medicaid programs to cover more low-income residents. And many declined to take responsibility for the online marketplaces — called exchanges— that would offer subsidized private coverage to the middle class.

Both would pose costs to the states and also involved cooperating with a larger government role in health care that many Republicans strongly opposed.

However, the federal government’s agreement to pay most of the added Medicaid expense, and belief that fewer residents would be showing up at local hospitals without insurance, have begun to break down some governors’ opposition.